By staying in position, he can also argue that his presence in government is a demonstration of its credibility on Brexit to the party's nervous right wing; if he were to depart in the future, it would be a signal that May has finally lost the confidence of her MPs.
Boris Johnson (file) in March.
Johnson´s dramatic resignation on Monday just hours after Brexit minister David Davis quit late on Sunday plunged the value of the pound on currency markets. Brexit-supporting lawmakers were angered by the proposals, saying they would keep Britain tethered to the bloc and unable to change its rules to strike new trade deals around the world.
Mrs May said she was "sorry - and a little surprised" by Mr Johnson's move after his apparent support on Friday.
The prime minister's official spokesman said she would fight any attempt to oust her as prime minister if the required 48 Tory MPs called for a contest.
After two hours in the Commons defending her plans yesterday afternoon, the PM faced Tory MPs behind closed doors - with many warning her offer gave too much away.
But ministerial aide Chris Green resigned his position as a parliamentary private secretary to the Department for Transport after the meeting, saying in a letter to Mrs May that she had confirmed his fears that "we would not really leave the EU" under her proposals.
The Brexit "dream is dying, suffocated by needless self-doubt", he said in the letter, which he posted to Twitter.
Both Brexiteers and Remainers might consider voting the deal down during the meaningful vote, with each side convinced that such an act would or wouldn't trigger a no-deal Brexit, respectively.
Hunt supported Remain during the 2016 referendum on EU membership, but has since been critical of the European Commission's approach to negotiations.
This morning Theresa May chairs cabinet.
Earlier, May defended her Brexit plan to lawmakers in the House of Commons, with Johnson absent from his usual place on the Conservative front bench.
Why did David Davis resign?
European Council President Donald Tusk said Monday that "the mess caused by Brexit is the biggest problem in the history of EU-UK relations and it is still very far from being resolved".
"It seems to me we're giving too much away, too easily, and that's a risky strategy at this time", Davis said in a BBC radio interview Monday morning.
Hunt, the MP for South West Surrey, has served as Health Secretary since 2012. If Labour were to signal it would definitely do the same, that could, in theory, leave May with a more serious problem: a looming veto over the divorce talks leaving the prime minister in an extremely hard position.
But she told MPs: "We do not agree on the best way to deliver our shared commitments to honour the result of the referendum".
May's proposals for a future European Union relationship after Britain departs from the bloc next March had taken two years of internal government wrangling to agree.
The majority of MPs are in favour of preserving frictionless trade across the UK-EU border and the open border on the island of Ireland.